Problem: You have found the following ages (in years) of all 5 sloths at your local zoo: $ 24,\enspace 15,\enspace 1,\enspace 3,\enspace 12$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Explanation: Because we have data for all 5 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{24 + 15 + 1 + 3 + 12}{{5}} = {11\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $24$ years $13$ years $169$ years $^2$ $15$ years $4$ years $16$ years $^2$ $1$ year $-10$ years $100$ years $^2$ $3$ years $-8$ years $64$ years $^2$ $12$ years $1$ year $1$ year $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{169} + {16} + {100} + {64} + {1}} {{5}} $ $ {\sigma^2} = \dfrac{{350}}{{5}} = {70\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{70\text{ years}^2}} = {8.4\text{ years}} $ The average sloth at the zoo is 11 years old. There is a standard deviation of 8.4 years.